Theorem rossspw 34193

Description: A ring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)

Hypothesis

Ref	Expression
isros.1	⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}

Assertion

Ref	Expression
rossspw	⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)

Distinct variable groups:   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦

Allowed substitution hints:   𝑄(𝑥,𝑦,𝑠)   𝑂(𝑥,𝑦)

Proof of Theorem rossspw

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isros.1	. . . 4 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}
2	1	isros 34192	. . 3 ⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)))
3	2	simp1bi 1145	. 2 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝒫 𝒫 𝑂)
4	3	elpwid 4560	1 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  {crab 3397   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  ∅c0 4284  𝒫 cpw 4551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-pw 4553

This theorem is referenced by:  rossros  34204